Spin variable

Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction... 

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

Slightly more complex models treat the water, the amphiphile and the oil as tliree distinct variables corresponding to the spin variables. S = +1, 0, and -1. The most general Hamiltonian with nearest-neighboiir interactions has the fomi... 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function boundary conditions that the ( /j obey can be expanded in this complete set... 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

So far, I have ignored the existenee of spin. Spin is an internal angular momentum that some partieles have and others do not. Eleetron spin is a two-valued quantity vve denote the spin variable for a single eleetron s, and the spin states are written o (s) and 3(s), or just a and p for short when the meaning is obvious. The notation I am going to use is that afsj) means eleetron 1 in spin state a. With an eye to the discussion above about indistinguishability, we consider the following four combinations of spin states for two electrons ... 

Here the integration J dr is over the coordinates of both electrons. Such integrals are therefore eight-dimensional (three spatial variables and one spin variable per electron). Integration over the spin variables is straightforward, but the spatial variables are far from easy a particular source of trouble arises from the electron repulsion term. 

Some authors write x = r s to denote the total variables of the electron, and write the total wavefunction as k(x) or F(r, s). 1 have used a capital here to emphasize that the total wavefunction depends on both the space and spin variables. I will use the symbol dr to denote a differential space element, and ds to denote a differential spin element. 

Many physical properties such as the electrostatic potential, the dipole moment and so on, do not depend on electron spin and so we can ask a slightly different question what is the chance that we will find the electron in a certain region of space dr irrespective of spin To find the answer, we integrate over the spin variable, and to use the example 5.2 above... 

There are 16 electrons in total, 14 cr and 2 n and so our total wavefunction will be some complicated function of their spatial (ri, rj,..., rie) and spin variables (si, S2,..., sie). The electronic state wavefimction for the molecule can therefore be written... 

Aj[ the beginning of this chapter, I introduced the notion that the 16 electrons iU ethene could be divided conceptually into two sets, the 14 a and the 2 n electrons. Let me refer to the space and spin variables as xi, Xj, > xi6, and for the minute I will formally label electrons 1 and 2 as the 7r-electrons, with 3 through 16 the cr-electrons. Methods such as Huckel rr-electron theory aim to treat the TT-electrons in an effective field due to the nuclei and the remaining a electrons. To see how this might be done, let s look at the electronic Hamiltonian end see if it can be sensibly partitioned into a rr-electron part (electrons 1 and 2) and a cr part (electrons 3 through 16). We have... 

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

The integration is over all the space and spin coordinates of electrons 2, 3,..., m. Many of the operators that represent physical properties do not depend on spin, and so we often average-out over the spin variable when dealing with such properties. The chance of finding electron 1 in the differential space element dt] with either spin, and the remaining electrons anywhere and with either spin is... 

Here, the integration is now over the spin variable for electron 1, as well as the space and spin variables for all the remaining electrons. 

In this section I will write h for the time-dependent states and ijr for the time-independent ones. The jri may themselves depend on the space and spin variables of all the particles present. 

The integral on the right-hand side is over space and spin variables, and so can be written... 

Don t confuse this with my earlier use of x for a space-spin variable the notation is common usage in both applications.) The Klein-Gordon equation is therefore... 

It is instructive to observe that in terms of conventional spin variables Si = 2ai - 1 (si = 1), and written in slightly more general form, takes on the form of the familiar Ising energy... 

We then say that the particle has spin and the three components Sl constitute the (pseudovector) spin operator. Note that by virtue of Eq. (9-55) the spin variables are not expressible in terms of the variables q and p. Since the angular momentum variables J are also the infinitesimal generators of rotations we deduce that... 

It is necessary to substitute a phenomenological equation for which in this case involves only spin variables. 

First we define the linear map that produces the densities from N-particle states. It is a map from the space of A -particle Trace Class operators into the space of complex valued absolute integrable functions of space-spin variables... 

Another class of expansions is also possible, but in these the functions cannot be interpreted as belonging to the Hilbert space of one-particle states, even though they are functions of one space and one spin variable and do belong to a Hilbert space. In such expansions the nwms of the functions are less than or equal to N and 1 < M < °°, while the trace of the density is equal to N. In the extreme case of Af = 1 one can even express the density as p = am, where... 

It must be underlined that, in (13), while the resulting RDM, pD, may be represented either in an orbital basis or in a spin-orbital one, as in (Ref. 17), the symbols A, fl stand for uniquely defined states depending on both space and spin variables. Relation (13) allows us to contract any q - RDM and what is more, it also allows us to contract any q - HRDM by replacing the number N by the number 2K -N). The derivation of the MCM is based on the important and well known relation... 

Equations (45) and (49) stress the direct connexion existing between the elements and classes of the Symmetric Group of Permutations and the terms derived by com-muting/anticommuting groups of fermion operators after summing with respect to the spin variables. 

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